{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "a8df7bfc",
   "metadata": {},
   "source": [
    "# 微积分"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "77f4763f",
   "metadata": {},
   "source": [
    "## 极限\n",
    "\n",
    "1. 极限的概念：设函数$f(x)$在点$a$处的导数存在，$f(x)$在$a$处可导，则$f(x)$在$a$处的极限为：\n",
    "\n",
    "$$\\lim_{x\\to a}f(x)=L$$\n",
    "\n",
    "当$x$趋于$a$时，$f(x)$趋于$L$，记作$f(x)\\to L$或$f(x)\\rightarrow L$。\n",
    "\n",
    "2. 极限的性质：\n",
    "\n",
    "- 2.1. 存在性：若$f(x)$在$a$处的导数存在，则$f(x)$在$a$处有极限。\n",
    "\n",
    "- 2.2. 唯一性：若$f(x)$在$a$处的导数存在，且$f(x)$在$a$处可导，则$f(x)$在$a$处只有唯一的极限$L$。\n",
    "\n",
    "- 2.3. 收敛性：若$f(x)$在$a$处的导数存在，且$f(x)$在$a$处可导，则$f(x)$在$a$处收敛当且仅当$\\forall \\epsilon>0,\\exists \\delta>0: |x-a|<\\delta \\Rightarrow |f(x)-L|<\\epsilon$。\n",
    "\n",
    "- 2.4. 连续性：若$f(x)$在$a$处的导数存在，且$f(x)$在$a$处可导，则$f(x)$在$a$处连续当且仅当$\\forall \\epsilon>0,\\exists \\delta>0: |x-a|<\\delta \\Rightarrow |f(x)-f(a)|<\\epsilon$。\n",
    "\n",
    "- 2.5. 夹逼定理：若$f(x)$在$a$处的导数存在，且$f(x)$在$a$处可导，则$f(x)$在$a$处的极限$L$存在，且$f(x)$在$a$处的导数$f'(a)$存在，则$f(x)$在$a$处的极限$L$等于$f(a)+f'(a)(x-a)$。\n",
    "\n",
    "- 2.6. 线性性：若$f(x)$在$a$处的导数存在，且$f(x)$在$a$处可导，则$f(x)$在$a$处的导数$f'(a)$存在，且$f'(a)\\neq 0$，则$f(x)$在$a$处的导数$f'(x)$存在，且$f'(x)=f'(a)+(x-a)f''(a)$。\n",
    "\n",
    "- 2.7. 鞅定理：若$f(x)$在$a$处的导数存在，且$f(x)$在$a$处可导，则$f(x)$在$a$处的导数$f'(a)$存在，且$f'(a)\\neq 0$，则$f(x)$在$a$处的极限$L$等于$f(a)+f'(a)\\frac{x-a}{f'(a)}$。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d11fa211",
   "metadata": {},
   "source": [
    "## 导数\n",
    "\n",
    "1. 导数的概念：设函数$f(x)$在点$a$处的导数$f'(a)$存在，则$f'(a)$称为$f(x)$在$a$处的导数。\n",
    "\n",
    "2. 导数的性质：\n",
    "\n",
    "- 2.1. 定义域：若$f(x)$在$a$处可导，则$f'(a)$在$a$处定义。\n",
    "\n",
    "- 2.2. 连续性：若$f(x)$在$a$处可导，则$f'(x)$在$a$处连续。\n",
    "\n",
    "- 2.3. 唯一性：若$f(x)$在$a$处可导，则$f'(x)$在$a$处唯一。\n",
    "\n",
    "- 2.4. 线性性：若$f(x)$在$a$处可导，则$f'(x)$在$a$处为线性函数。\n",
    "\n",
    "- 2.5. 泰勒公式：若$f(x)$在$a$处可导，则：\n",
    "\n",
    "> $$f(x)=f(a)+f'(a)(x-a)+\\frac{f''(a)}{2!}(x-a)^2+\\frac{f'''(a)}{3!}(x-a)^3+\\cdots+\\frac{f^{(n)}(a)}{n!}(x-a)^n+\\cdots$$\n",
    "> 当$x$趋于$a$时，$f(x)$趋于$f(a)$，且$f(x)$的$n$阶导数$f^{(n)}(x)$存在，则：\n",
    "> $$f(x)\\approx f(a)+f'(a)(x-a)+\\frac{f''(a)}{2!}(x-a)^2+\\frac{f'''(a)}{3!}(x-a)^3+\\cdots+\\frac{f^{(n)}(a)}{n!}(x-a)^n+\\cdots$$\n",
    "> 当$x$趋于$a$时，$f(x)$的$n$阶导数$f^{(n)}(x)$趋于$f^{(n)}(a)$。\n",
    "\n",
    "- 2.6. 牛顿-莱布尼兹公式：若$f(x)$在$a$处可导，则：\n",
    "\n",
    "> $$f(x)=\\frac{f(b)-f(a)}{x-a}+\\frac{1}{2!}(x-a)^2f''(a)+\\frac{1}{3!}(x-a)^3f'''(a)+\\cdots+\\frac{1}{n!}(x-a)^nf^{(n)}(a)+\\cdots$$\n",
    "> 当$x$趋于$a$时，$f(x)$趋于$f(a)$，且$f(x)$的$n$阶导数$f^{(n)}(x)$存在，则：\n",
    "> $$f(x)\\approx \\frac{f(b)-f(a)}{x-a}+\\frac{1}{2!}(x-a)^2f''(a)+\\frac{1}{3!}(x-a)^3f'''(a)+\\cdots+\\frac{1}{n!}(x-a)^nf^{(n)}(a)+\\cdots$$\n",
    "> 当$x$趋于$a$时，$f(x)$的$n$阶导数$f^{(n)}(x)$趋于$f^{(n)}(a)$。\n",
    "\n",
    "- 2.7. 导数的求法：\n",
    "\n",
    "- 2.7.1. 利用公式：若$f(x)$在$a$处可导，则：\n",
    "\n",
    "> $$f'(x)=\\lim_{h\\to 0}\\frac{f(x+h)-f(x)}{h}$$\n",
    "> 当$h$趋于$0$时，$f'(x)$趋于$f'(a)$。\n",
    "\n",
    "- 2.7.2. 利用微分法：若$f(x)$在$a$处可导，则：\n",
    "\n",
    "> $$f'(x)=\\frac{d}{dx}f(x)$$\n",
    "> 利用微分法求导，得到：\n",
    "> $$f'(x)=\\frac{d}{dx}f(x)=\\frac{f(x+h)-f(x)}{h}$$\n",
    "> 当$h$趋于$0$时，$f'(x)$趋于$f'(a)$。\n",
    "\n",
    "- 2.7.3. 利用导数的定义：若$f(x)$在$a$处可导，则：\n",
    "\n",
    "> $$f'(a)=\\lim_{x\\to a}\\frac{f(x)-f(a)}{x-a}$$\n",
    "> 当$x$趋于$a$时，$f'(x)$趋于$f'(a)$。\n",
    "\n",
    "- 2.7.4. 利用泰勒公式：若$f(x)$在$a$处可导，则：\n",
    "\n",
    "> $$f'(x)=\\frac{f(x+h)-f(x)}{h}=\\frac{f(x+h)-f(a)-f'(a)(h)}{h}+\\frac{f'(a)}{h}$$\n",
    "> 当$h$趋于$0$时，$f'(x)$趋于$f'(a)$。\n",
    "\n",
    "- 2.7.5. 利用牛顿-莱布尼兹公式：若$f(x)$在$a$处可导，则：\n",
    "\n",
    "> $$f'(x)=\\frac{f(b)-f(a)}{x-a}+\\frac{1}{2!}(x-a)^2f''(a)+\\frac{1}{3!}(x-a)^3f'''(a)+\\cdots+\\frac{1}{n!}(x-a)^nf^{(n)}(a)+\\cdots$$\n",
    "> 当$x$趋于$a$时，$f(x)$趋于$f(a)$，且$f(x)$的$n$阶导数$f^{(n)}(x)$存在，则：\n",
    "> $$f'(x)\\approx \\frac{f(b)-f(a)}{x-a}+\\frac{1}{2!}(x-a)^2f''(a)+\\frac{1}{3!}(x-a)^3f'''(a)+\\cdots+\\frac{1}{n!}(x-a)^nf^{(n)}(a)+\\cdots$$\n",
    "> 当$x$趋于$a$时，$f(x)$的$n$阶导数$f^{(n)}(x)$趋于$f^{(n)}(a)$。\t"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ade6ab6d",
   "metadata": {},
   "source": [
    "## 积分\n",
    "\n",
    "1. 积分的概念：设函数$f(x)$在区间$[a,b]$上连续，则$f(x)$在$a$处的积分$\\int_a^b f(t)dt$称为$f(x)$在$a$处的积分。\n",
    "\n",
    "2. 积分的性质：\n",
    "\n",
    "- 2.1. 定义域：若$f(x)$在$a$处连续，则$\\int_a^b f(t)dt$在$a$处定义。\n",
    "\n",
    "- 2.2. 线性性：若$f(x)$在$a$处连续，则$\\int_a^b f(t)dt$在$a$处为线性函数。\n",
    "\n",
    "- 2.3. 定积分：若$f(x)$在$a$处连续，则：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\lim_{n\\to\\infty}\\sum_{i=1}^n\\frac{b-a}{n}f(a+i\\frac{b-a}{n})$$\n",
    "\n",
    "当$n$趋于无穷大时，$\\int_a^b f(x)dx$趋于$f(b)$。\n",
    "\n",
    "- 2.4. 积分的求法：\n",
    "\n",
    "- 2.4.1. 利用公式：若$f(x)$在$a$处连续，则：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^c f(x)dx+\\int_c^b f(x)dx$$\n",
    "\n",
    "利用定积分的线性性，得到：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^c f(x)dx+\\int_c^b f(x)dx=\\int_a^c f(x)dx+\\int_a^c f(x)dx+\\int_c^b f(x)dx+\\int_c^b f(x)dx$$\n",
    "\n",
    "当$c$取$a$时，$\\int_a^c f(x)dx$为$0$，当$c$取$b$时，$\\int_c^b f(x)dx$为$0$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^c f(x)dx+\\int_c^b f(x)dx=\\int_a^c f(x)dx+\\int_a^b f(x)dx$$\n",
    "\n",
    "当$c$取$a$时，$\\int_a^b f(x)dx$为$f(b)$，因此：\t\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^c f(x)dx+\\int_a^b f(x)dx=f(b)$$\n",
    "\n",
    "当$c$取$b$时，$\\int_c^b f(x)dx$为$f(a)$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_c^b f(x)dx+\\int_a^b f(x)dx=f(a)$$\n",
    "\n",
    "- 2.4.2. 利用微积分公式：若$f(x)$在$a$处连续，则：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^b \\frac{d}{dx}f(x)dx+\\int_a^b f(a)dx$$\n",
    "\n",
    "利用微分法求导，得到：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^b \\frac{d}{dx}f(x)dx+\\int_a^b f(a)dx=\\int_a^b f'(x)dx+\\int_a^b f(a)dx$$\n",
    "\n",
    "当$x$取$a$时，$\\int_a^b f'(x)dx$为$f(a)$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^b f'(x)dx+\\int_a^b f(a)dx=f(a)$$\n",
    "\n",
    "当$x$取$b$时，$\\int_a^b f'(x)dx$为$f(b)$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^b f'(x)dx+\\int_a^b f(b)dx=f(b)$$\n",
    "\n",
    "- 2.4.3. 利用积分恒等式：若$f(x)$在$a$处连续，则：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^b f(t)dt$$\n",
    "\n",
    "利用定积分的线性性，得到：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^b f(t)dt=\\int_a^c f(t)dt+\\int_c^b f(t)dt$$\n",
    "\n",
    "当$c$取$a$时，$\\int_a^c f(t)dt$为$0$，当$c$取$b$时，$\\int_c^b f(t)dt$为$0$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^c f(t)dt+\\int_c^b f(t)dt=\\int_a^b f(t)dt$$\n",
    "\n",
    "当$c$取$a$时，$\\int_a^b f(t)dt$为$f(b)$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^b f(t)dt=f(b)$$\n",
    "\n",
    "当$c$取$b$时，$\\int_c^b f(t)dt$为$f(a)$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_c^b f(t)dt=f(a)$$\n",
    "\n",
    "- 2.4.4. 利用积分公式：若$f(x)$在$a$处连续，则：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^c f(x)dx+\\int_c^b f(x)dx=\\int_a^c f(t)dt+\\int_c^b f(t)dt$$\n",
    "\n",
    "利用积分恒等式，得到：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^c f(t)dt+\\int_c^b f(t)dt=\\int_a^c f(t)dt+\\int_a^b f(t)dt+\\int_c^b f(t)dt+\\int_c^b f(t)dt$$\n",
    "\n",
    "当$c$取$a$时，$\\int_a^c f(t)dt$为$f(b)$，当$c$取$b$时，$\\int_c^b f(t)dt$为$f(a)$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^c f(t)dt+\\int_a^b f(t)dt+\\int_c^b f(t)dt+\\int_c^b f(t)dt=f(b)+f(a)+f(b)+f(a)$$\n",
    "\n",
    "当$c$取$a$时，$\\int_a^b f(t)dt$为$f(b)$，当$c$取$b$时，$\\int_c^b f(t)dt$为$f(a)$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^b f(t)dt=f(b)+f(a)$$\n",
    "\n",
    "当$c$取$a$时，$\\int_a^b f(t)dt$为$f(b)$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_a^b f(t)dt=f(b)$$\n",
    "\n",
    "当$c$取$b$时，$\\int_c^b f(t)dt$为$f(a)$，因此：\n",
    "\n",
    "$$\\int_a^b f(x)dx=\\int_c^b f(t)dt=f(a)$$\t"
   ]
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